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    The Dirichlet problem for elliptic systems with data in Köthe function spaces

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    Genre
    Journal Article
    Date
    2016-01-01
    Author
    Martell, JM
    Mitrea, D
    Mitrea, I
    Mitrea, M
    Subject
    Dirichlet problem
    second-order elliptic system
    nontangential maximal function
    Hardy-Littlewood maximal operator
    Poisson kernel
    Green function
    Kothe function space
    Muckenhoupt weight
    Lebesgue space
    variable exponent Lebesgue space
    Lorentz space
    Zygmund space
    Orlicz space
    Hardy space
    Beurling algebra
    Hardy-Beurling space
    semigroup
    Fatou type theorem
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    Permanent link to this record
    http://hdl.handle.net/20.500.12613/5746
    
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    DOI
    10.4171/rmi/903
    Abstract
    © 2016 European Mathematical Society. We show that the boundedness of the Hardy-Littlewood maximal operator on a Kothe function space X and on its Kothe dual X is equivalent to the well-posedness of the X-Dirichlet and X-Dirichlet problems in Rn + in the class of all second-order, homogeneous, elliptic systems, with constant complex coefficients. As a consequence, we obtain that the Dirichlet problem for such systems is well-posed for boundary data in Lebesgue spaces, variable exponent Lebesgue spaces, Lorentz spaces, Zygmund spaces, as well as their weighted versions. We also discuss a version of the aforementioned result which contains, as a particular case, the Dirichlet problem for elliptic systems with data in the classical Hardy space H1, and the Beurling-Hardy space HAp for p € (1,∞). Based on the well-posedness of the Lp-Dirichlet problem we then prove the uniqueness of the Poisson kernel associated with such systems, as well as the fact that they generate a strongly continuous semigroup in natural settings. Finally, we establish a general Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for null-solutions of such systems.
    Citation to related work
    European Mathematical Society Publishing House
    Has part
    Revista Matematica Iberoamericana
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    For Americans with Disabilities Act (ADA) accommodation, including help with reading this content, please contact scholarshare@temple.edu
    ae974a485f413a2113503eed53cd6c53
    http://dx.doi.org/10.34944/dspace/5728
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